In a regular triangular pyramid SABC with base ABC on the median CE, point K is taken so that CK: KE

| May 24, 2021 | education

In a regular triangular pyramid SABC with base ABC on the median CE, point K is taken so that CK: KE = 8: 1. A plane is drawn through the point K, perpendicular to the straight line CE and intersects the lateral edges SA and SB at points M and N, respectively. Prove that MN: AB = 2: 3.

Empty median length CE is 9 * X cm.

Then, by condition, the length of the segment CK = 8 * X cm, and KE = X cm.

By the property of the median of the triangle, it is divided by the ratio of 2/1 at the point of their intersection.

Then CO = 6 * X cm, OE = 3 * X cm.

Then OK = CK – CO = 8 * X – 6 * X = 2 * X cm.

The ratio OK / OE = 2 * X / 3 * X = 2/3.

Let’s build the SE apothem. Right-angled triangles SOE and RKE are similar in acute angle and the coefficient of their similarity is OK / OE = 2/3. Then SE / SP = 2/3.

The segment MH is parallel to AB, then the triangles SAB and SMH are similar in two angles, and the segments SP and SE are their heights. Since SE / SP = 2/3, then MH / AB = 2/3, as required.



One of the components of a person's success in our time is receiving modern high-quality education, mastering the knowledge, skills and abilities necessary for life in society. A person today needs to study almost all his life, mastering everything new and new, acquiring the necessary professional qualities.